Riley, while walking to school on Monday morning, observes a plane flying directly overhead at an altitude of 5km.
The diagram shows the angle of elevation to the plane, θ, and the horizontal distance, D, from Riley.
a. Show that the horizontal distance
D = 5/tanθ
b. Hence, show that the rate of change of the horizontal distance, in simplest form, is given by:
dD/dt = -5/sin^2θ dθ/dt
c. Given that the plane is moving at a constant speed of 780km/h, find the rate at which the angle of elevation is changing at the instant when θ = π/6 and interpret your answer.
Answers
a. D = 5/tan(θ)
b. dD/dt = -5/sin^2(θ) * dθ/dt
c. At θ = π/6, the rate at which the angle of elevation is changing is -78 km/h, indicating the plane is descending.
To solve this problem, we'll start by using trigonometry to relate the given information and then differentiate the equation to find the rate of change.
a. Let's consider the right triangle formed by the plane, Riley, and the point on the ground directly beneath the plane. The horizontal distance from Riley to that point is D, and the vertical distance (altitude of the plane) is 5 km. The angle of elevation to the plane is θ.
Using trigonometry, we have:
tan(θ) = (opposite side) / (adjacent side)
tan(θ) = 5 / D
Rearranging this equation, we get:
D = 5 / tan(θ)
b. To find the rate of change of the horizontal distance, we need to differentiate the equation with respect to time (t). Let's denote the rate of change of D as dD/dt and the rate of change of θ as dθ/dt.
Differentiating both sides of the equation D = 5 / tan(θ) with respect to t, we get:
dD/dt = d(5/tan(θ))/dt
Using the quotient rule for differentiation, we have:
dD/dt = (-5 sec^2(θ) dθ/dt) / (tan^2(θ))
Recall that sec^2(θ) = 1 + tan^2(θ), so we can rewrite the equation as:
dD/dt = (-5 dθ/dt) / (tan^2(θ)) * (1 + tan^2(θ))
Simplifying further, we get:
dD/dt = -5 dθ/dt / (sin^2(θ))
c. Given that the plane is moving at a constant speed of 780 km/h, we need to find the rate at which the angle of elevation is changing (dθ/dt) at the instant when θ = π/6.
Substituting θ = π/6 into the equation from part b, we have:
dD/dt = -5 dθ/dt / (sin^2(π/6))
Since sin(π/6) = 1/2, we can simplify the equation to:
dD/dt = -10 dθ/dt
We know that the speed of the plane is constant at 780 km/h, which means the horizontal distance (D) is changing at a constant rate. Therefore, dD/dt = 780 km/h.
Substituting this value into the equation, we have:
780 km/h = -10 dθ/dt
Solving for dθ/dt, we get:
dθ/dt = -780 km/h / 10
dθ/dt = -78 km/h
Interpretation: The rate at which the angle of elevation is changing at the instant when θ = π/6 is -78 km/h. This negative sign indicates that the angle is decreasing. Therefore, the plane is descending at an angle of approximately -78 km/h.
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Related Questions
f(x)=x⁵+5x⁴+10x²−1
local minimum value of ___ at x = ___
local maximum value of ___ at x = ____
Increase: ___
Decrease: ___
Answers
The local minimum value of the function f(x) = \(x^{5}\) + 5\(x^{4}\)+ 10\(x^{2}\) - 1 occurs at x = -1, with a value of -6. The local maximum value of the function occurs at x = 0, with a value of -1. As x approaches negative or positive infinity, the function increases without bound, and as x approaches negative or positive infinity, the function decreases without bound.
To find the local minimum and maximum values of the function, we can take the derivative of the function and set it equal to zero to find the critical points. Taking the derivative of f(x) with respect to x gives us f'(x) = 5\(x^{4}\)+ 20\(x^{3}\) + 20x. Setting f'(x) equal to zero and solving for x, we find that x = -1 and x = 0 are the critical points.
To determine whether these critical points are local minimum or maximum points, we can use the second derivative test. Taking the second derivative of f(x) gives us f''(x) = 20\(x^{3}\) + 60\(x^{2}\)+ 20. Evaluating f''(x) at x = -1 and x = 0, we find that f''(-1) = 100 and f''(0) = 20. Since f''(-1) > 0 and f''(0) > 0, we can conclude that x = -1 is a local minimum and x = 0 is a local maximum.
Therefore, the local minimum value of the function f(x) = \(x^{5}\) + 5\(x^{4}\) + 10\(x^{2}\) - 1 is -6 at x = -1, and the local maximum value is -1 at x = 0. The function increases without bound as x approaches negative or positive infinity, and it decreases without bound as x approaches negative or positive infinity.
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||3x-4|-5|>1
pls solve i need help with it i give 100 points
Answers
Answer:
\({ \tt{ | |3x - 4| - 5 | > 1 }} \\ \\ { \tt{ |3x - 4| > 1 ±5}}\)
• Either |3x - 4| > 1 + 5 or |3x - 4| > 1 - 5
\({ \tt{ |3x - 4| > 6 \: \: or \: \: - 4}}\)
→ for |3x - 4| > 6:
\({ \tt{ |3x - 4| > 6}} \\ \\ { \tt{3x > 6±4}} \\ \\ { \boxed{ \tt{ \: x = \frac{10}{3} \: \: or \: \: \frac{2}{3} }}}\)
→ for |3x - 4| > -4:
\({ \tt{ |3x - 4| > - 4}} \\ \\ { \tt{3x > - 4±4}} \\ \\ { \boxed{ \tt{ \: x > 0 \: \: or \: \: - \frac{8}{3} }}}\)
n C (x) = 0.6x² − 168x+17,507. What is the minimum unit cost?
-
Answers
Step-by-step explanation:
Answer:
Minimum Unit Cost = $14,362
Step-by-step explanation:
The standard form of a quadratic is given by:
ax^2 + bx + c
So for our function, we can say,
a = 0.6
b = -108
c = 19,222
We can find the vertex (x-coordinate where minimum value occurs) by the formula -b/2a
So,
-(-108)/2(0.6) = 108/1.2 = 90
Plugging this value into original function would give us the minimum (unit cost):
how to find the equation of a quadratic function given 3 points
Answers
1. Write the general form of a quadratic function: y = ax^2 + bx + c
2. Plug in the coordinates of the first point to get an equation: y1 = a(x1)^2 + b(x1) + c
3. Plug in the coordinates of the second point to get another equation: y2 = a(x2)^2 + b(x2) + c
4. Plug in the coordinates of the third point to get a third equation: y3 = a(x3)^2 + b(x3) + c
5. Use a system of equations to solve for a, b, and c. You can do this by either substitution or elimination.
Once you have solved for a, b, and c, you can plug them back into the general form of the quadratic function to get the equation.
Which of the following values is not an irrational number? 5.73 repeating where the 3 is repeating, pi, the cube root of 73, and the square root of 40 5.73 repeating where the 3 is repeating π the cube root of 73 the square root of 40
Answers
Answer:
A
Step-by-step explanation:
5.73 with 3 repeating because can be expressed as a fraction
(573-57)/90 = 516/90
the value of “y” varies directly with “x”. if y= 56, then x= 4
Answers
there are 6 students in the class. one of them is to be selected as the best student and another student is to be selected as a runner-up. how many different ways can this be done?
Answers
If there are 6 students and one is selected for best student and another is selected for runner up, then 15 different ways that we can arrange the students
Total number of students in the class = 6 students
Number of students for best student = 1 student
Number of students for runner up = 1 students
Total number of students needed = 2
Here we have to use the combination
6\(C_2\) = 6! / 2!(6 - 2)!
= 6! / 2! × 4!
Split the terms and cancel it
= 6 × 5 / 2 × 1
Multiply the terms
= 30 / 2
Divide the numbers
= 15 combinations
Therefore, there are 15 combinations
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What are the period, phase shift, and vertical shift of y = csc[3(x 4)] 6?
Answers
The period of the given function is 2π/3, the phase shift is 4, and the vertical shift is 6 units.
What is a function?It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.
We have a trigonometric function:
y = csc[3(x+ 4)]+ 6
Every each revolution of 2π the function repeats the same value for every x.
The period of the original function is 2π
For the given function plug 2π
y = csc[3(x+ 4)+2π]+ 6
y = csc[3(x+ 2π/3+4)]+ 6
The period of the function will be 2π/3
From the function, the phase shift = 4
And vertical shift = 6
Thus, the period of the given function is 2π/3, the phase shift is 4, and the vertical shift is 6 units.
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Option 3
Step-by-step explanation:
PLEASE HELP!!!! ASAP!!!!!
What matrix is the result of m x h?
Fill in the boxes. Enter any fractions as simplified fractions.
M x H [ _ _ _]
Answers
Answer:
The correct answer is -9, 36, -9/2
Step-by-step explanation:
If 5x-3=3x+5, what is the value of 2X-8? 5
Answers
Answer:
5x-3=3x+5
5x-3x=5+3
2x=8
x=8/2
x=4
Therefore,
Value of 2x-8
Substitute the value of x
2(4)-8
8-8=0
If you are helped with my answer pls tag me brianliest
I really want it
Plss
THANK you.....
100-475.50<1200 slove
Answers
Answer:
-375.5<1200
True
Step-by-step explanation:
100-475.5 = -375.5
Brainliest please!!
Please help :)
The total distance was 540 miles. George drove part way going 40mph on a motorcycle and part way going 60 mph in a car. If the total time was 11 hours, what distance did George cover in the motorcycle?
Answers
Answer:
6 hours
Step-by-step explanation:
total: 540 mi
drove motorcycle: 40mph
drove car: 60 mph
11 hours
equation for miles: 540=40x+60y
equation for time: 11=x+y
11=x+y
-x -x
y=-x+11 <---- plug in
540=40x+60(-x+11)
540=40x-60x+660
-660 -660
-120=-20x
/-20 /-20
x=6
6 hours
In a class full of men and women,2/9
of the class are women. What is the ratio of men to
women in its simplest form?
Answers
Answer:
the ratio of men to women is 7 to 2
Step-by-step explanation:
you have 9 people 7 are men and 2 are women
Answer: this is kinda already the simplest form unless you want to do 1/4.5
Step-by-step explanation:
what is the slope y=5x-3
Answers
Answer:
The slope is 5.
Explanation:
In the slope-intercept form of an equation
\(y=mx+b\)m is the slope, and b is the y-intercept of the equation.
Now in our case,
\(\begin{gathered} m=5 \\ b=-3 \end{gathered}\)therefore, the slope is 5 and the y-intercept is -3.
What is the equation of this graphed line
Answers
The equation of the graphed line is written as: y = 2x + 2.
How to Find the Equation of a Graphed Line?The equation of a graphed line can be determined by calculating the slope of the line, m = rise/run, and determining the y-intercept value, b, which is the point at which the line intersects the y-axis.
Using two points on the graphed line in the diagram attached below, (-4, -6) and (2, 6):
Slope (m) = change in y / change in x = (-6 - 6) /(-4 - 2)
Slope (m) = -12/-6
Slope (m) = 2
The line cuts the y-axis at y = 2. The y-intercept of the line, b = 2.
Wrote the equation of the graphed line by substituting m = 2 and b = 2 into y = mx + b:
y = 2x + 2.
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Examine the words and/or phrases below and determine the relationship among the majority of words/phrases. Choose the option which does not fit the pattern.
horn
arête
drumlin
lateral moraine
Answers
The majority of the words/phrases are related to glacial features or landforms. They represent various aspects of glacial processes and landform formation. However, "drumlin" does not fit this pattern as it specifically refers to a type of glacial landform.
Among the words/phrases provided, the majority are geological features or landforms. "Horn," "arête," "drumlin," and "lateral moraine" are all specific terms used in geology to describe different land formations.
A "horn" is a pointed mountain peak formed by the erosion of glaciers from several sides. An "arête" is a narrow ridge that separates two adjacent glacial valleys. A "drumlin" is an elongated hill formed by glacial deposition and erosion. A "lateral moraine" is a ridge of debris deposited along the sides of a glacier.
These words are related to glacial processes and landforms, specifically. They all represent different features associated with glacial erosion, deposition, or the formation of glacial landforms.
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If Karri takes a 7 question true false test and guess on every question, what is the
probability that she gets all 7 questions correct?
Answers
Answer:
The probability that she gets all 7 questions correct is 0.0078.
Step-by-step explanation:
We are given that Karri takes a 7 question true-false test and guess on every question.
The above situation can be represented through binomial distribution;
\(P(X = r) = \binom{n}{r} \times p^{r} \times (1-p)^{n-r} ; x = 0,1,2,3,........\)
where, n = number of samples (trials) taken = 7 question
r = number of success = all 7 questions correct
p = probability of success which in our question is probability that
question is correct, i.e. p = 50%
Let X = Number of question that are correct
So, X ~ Binom(n = 7, p = 0.50)
Now, the probability that she gets all 7 questions correct is given by = P(X = 7)
P(X = 7) = \(\binom{7}{7} \times 0.50^{7} \times (1-0.50)^{7-7}\)
= \(1 \times 0.50^{7} \times 0.50^{0}\)
= 0.0078
Calculate the relative frequency of the data to determine which association the two-way table suggests.
A. None of the associations listed are correct.
B. Those who have a brother tend not to have a sister.
C. Those who have a brother tend to have a sister.
D. Those who do not have a brother tend not to have a sister.
Answers
The correct option A, "None of the associations listed are correct," is the appropriate response.To determine the association suggested by the two-way table, we need to calculate the relative frequency of the data.
The table provides information about whether individuals have a brother and a sister.
Based on the options given, let's calculate the relative frequencies to see which association is suggested:
Calculate the relative frequency for individuals who have a brother and a sister:Relative frequency = (Number of individuals with both a brother and a sister) / (Total number of individuals)
Calculate the relative frequency for individuals who have a brother but no sister:Relative frequency = (Number of individuals with a brother but no sister) / (Total number of individuals)
Calculate the relative frequency for individuals who have a sister but no brother:Relative frequency = (Number of individuals with a sister but no brother) / (Total number of individuals)
Calculate the relative frequency for individuals who have neither a brother nor a sister:Relative frequency = (Number of individuals with neither a brother nor a sister) / (Total number of individuals)
By comparing the relative frequencies, we can determine which association is suggested by the data.Unfortunately, the given two-way table is missing, so we cannot perform the necessary calculations to determine the association.
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1. COORDINATE GEOMETRY Classily AABC with vertices A-2, -1), B(-1,3), and C(2,0) as scalene, equilateral or isosceles.
Answers
Answer:
I'm pretty sure its a equilateral
Step-by-step explanation:
I graphed it
I really need help with this
Answers
Answer:
x= 10
Step-by-step explanation:
The angles are corresponding angles, and if corresponding angles are equal, the lines are parallel
4x = 3x+10
Subtract 3x from each side
4x-3x = 3x+10-10
x = 10
PLEASE HELP At the same farmer's market there was a fruit stand. The fruit stand sold a
bushel of apples for $12.00 and a bushel of pears for $11.00. One man ordered 2
bushels of apples and his bill came to $57.00. How many bushels of pears did he
order? USE THE EQUATION: 12a + 11p = 57
Answers
Answer:
3 bushels of pears
Step-by-step explanation:
$12x2=$24 $57-$24=$33
$33/$11=3
(I don’t understand how I would use the equation you gave, it doesn’t make sense)
or each type of effect listed—main effects, two-way interactions, and three-way interactions—identify the maximum number of possible effects that could be tested in a 2 × 2 × 2 factorial desig
Answers
Therefore, in a 2 × 2 × 2 factorial design, the maximum number of effects that could be tested are 3 main effects, 3 two-way interactions, and 1 three-way interaction.
In a 2 × 2 × 2 factorial design, the maximum number of possible effects that could be tested for each type are as follows:
Main Effects: There are three main effects, one for each factor (two levels per factor). So, the maximum number of main effects is 3.
Two-Way Interactions: There are three possible two-way interactions, one for each pair of factors. So, the maximum number of two-way interactions is 3.
Three-Way Interactions: There is only one three-way interaction, which involves all three factors. So, the maximum number of three-way interactions is 1.
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Which equation represents a population of 490 animals that decreases at an annual rate of 20% ?
Answers
The equation which represents a population of 490 animals that decreases at an annual rate of 20% is;
\(P(x)=490(1+0.20)^{x}\)
What is exponential function?When the input variable x appears as an exponent in the formula \(P(x)=a b^{x}\) the exponential function is indicated.
The exponential curve is influenced by both the exponential function and thus the value of x.
where, a = initial value.
b = 1 + r (rate of increase or decrease)
The animals' original population is indicated here by: 490
Thus, a = 490
Additionally, 20% is the rate of increase.
So, r = 20%
r = 0.20
As a result, the population function, or the number of animals in a population, is as follows:
\(P(x)=490(1+0.20)^{x}\)
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f(x) = 2x^2 Find f (-6)
Answers
Answer:
\(72\)
Step-by-step explanation:
\(f(x)=2x^2\)
replace x with -6
\(f(-6)=2(-6)^2\)
exponent
\(f(-6)=2(36)\)
multiply
\(f(-6)=72\)
Express 3.23×10 ^6
in standard decimal notation.
Answers
The number 3.23×10^6, expressed in standard decimal notation, is 3,230,000. In scientific notation, a number is expressed as the product of a coefficient and a power of 10.
In this case, the coefficient is 3.23, and the power of 10 is 6. To convert this number into standard decimal notation, we need to multiply the coefficient by the corresponding power of 10. The power of 10, in this case, is 10^6, which means 10 raised to the power of 6. This is equivalent to 1 followed by six zeros: 1,000,000. To obtain the standard decimal notation, we multiply the coefficient, 3.23, by 1,000,000.
Calculating the multiplication, we have:
3.23 × 1,000,000 = 3,230,000.
Hence, the number 3.23×10^6, expressed in standard decimal notation, is 3,230,000. This means that the number is equal to 3.23 million.
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Something's Strange about this little problem..
99 + 2 = ?
Answers
Answer:
the answer is 101 !!!
There's isn't nothing strange your just over thinking it!
Trust your gut, thanks for the points :)
Have a good night !!
hii I just wasn't sure about how to answer this so could someone help?
There are 2 bran, 3 blueberry and 4 corn muffins in a bag. If Wesley randomly selects two muffins, what is the probability that he selects a corn muffin, eats it, and then selects a bran muffin?
Answers
Answer:
1/2
Step-by-step explanation:
Answer:
1/9
Step-by-step explanation:
Bob is interested in examining the relationship between the number of bedrooms in a home and its selling price. After downloading a valid data set from the internet, he calculates the correlation. The correlation value he calculates is only 0.05What does Bob conclude? A)Bob gives up on his research becauser=.05 means there is no relationship of any kind between bedrooms and selling price. B) Bob continues his research because even though there is no linear relationship here, there could be a different relationship.
Answers
Bob continues his research because even though there is no linear equation relationship between the number of bedrooms and selling price, there could be another type of relationship.
Bob continues his research because even though the correlation value he calculated between the number of bedrooms and the selling price was only 0.05, this does not necessarily mean that there is no relationship of any kind between these two variables. It simply means that there is no linear relationship between the two. There could still be other types of relationships between the two variables, such as an exponential or quadratic relationship. It is possible that further analysis and exploration of the data set would reveal such a relationship and allow Bob to uncover the relationship between the two variables. Additionally, even if the correlation value is low, it does not necessarily mean that there is no relationship between the two variables. It simply means that the relationship is weak. Therefore, Bob should continue his research in order to uncover any potential relationships between the number of bedrooms and selling price.
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a trapezoid is a quadrilateral with exactly one pair of parallel sides. true or false?
Answers
it is going to be true
Which statement best describes how to determine whether f(x) = x² – X + 8 is an even function?
1. Determine whether -x²-(-x)+8 is equivalent to x²-x+8
2. Determine whether (x)²-x+8 is equivalent to x²-x+8
3. Determine whether -x²(-x)+8 is equivalent to -(x²-x+8)
4. Determine whether (-2)²-(-x)+8 is equivalent to -(x²-x+8)
Answers
Answer:
Option 1
Step-by-step explanation:
You can tell if a function is even if \(f(-x)=f(x)\):
\(f(x)=x^2-x+8\)
\(f(-x)=(-x)^2-(-x)+8\)
\(f(-x)=x^2+x+8\)
Since \(f(-x)\neq f(x)\), the function is not even. Therefore, option 1 is the best option.